I am once again trudging into the deep forest of how prime numbers, discrete semiprimes, and other composite numbers distribute themselves among the positive odd integers, looking for patterns. The illustrations below are from the beginning steps, taken yesterday and today, of this stage of my math study.

The table I’ve built using Apple Numbers spreads out most of the odd positive integers not greater than 675 positioned according to their representation m=c^{2}-r where c is the smallest integer greater than the square root of m that is odd or even as (m+1)/2 is odd or even. This is the method of deriving the Ceiling Square c that is consistent with the oddness or evenness of the larger square every representation of m as the difference of two perfect squares. Recall that an odd positive m greater than 1 has as many such representations as it has factorizations m=ab, with the greater perfect square being (a+b)/2 squared, and the lesser being (a-b)/2 squared. Remainders are congruent to 0 or 1 modulo 4, in harmony with the smaller square in all of m’s Fermat factorizations, and this table so far lists all such remainders less than 100, with c going up to 26. Some of the values less than 675 would actually appear in the row for c=27. It is a peculiar ordering of the odd positive integers in this way, but it serves the purposes of my study for the time being: spreading out the distribution of numbers according to Ceiling Square vs. remainder.

The whole table is too large to be readable on this page, so here is an image of the first few columns:

I colored odd perfect squares yellow, prime numbers green, and discrete semiprimes blue. For each representation of m as the product of two factors, in the cell corresponding to m the quantity I call the Ascent follows m. Since for all positive integers m equals 1 times m, all of the cells have at least (m+1)/2-c as their second number. For composite integers, the other Ascents (a+b)/2-c follow. These Ascents are the quantities I am seeking to find formulaically.

Notice that a few of the cells are purple. I did this for the table (but not for the “QR Code” below) since the cube of an odd prime, like 27, 125, and 343, has three (indistinct) prime factors but only two different representations m=ab. Thus I wanted to distinguish them from discrete semiprimes, which also have only two m=ab representations, and from the other composite numbers as well.

When r, the remainder in m=c^{2}-r, is a perfect square, we have the special situation that the Ceiling Square representation of m gives its Fermat factorization form. As long as r is not (c-1)^{2}, this will give a non-trivial factorization. So the corresponding columns in the table have a golden-yellow header background.

And last of all, I took the cells for integers that are not perfect squares and converted them into hexagons to pack them together, forming a honeycomb-like pattern (hey, remember my beehives and bee lines from some months ago?) that, when colored as the pertinent portion of the table is colored, becomes what I have begun to think of as part of Mathematics’ QR Code:

Software Applications Used: I produced the table with Apple Numbers, and the honeycomb chart with Virtual Graph Paper on their website. For calculations, I used my own get_divisors.pl Perl script, which outputs Ascents as well as all m=ab factorizations.

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